# The Topology of Statistical Verifiability

arXiv: https://arxiv.org/abs/1707.09378 · HTML: https://mlsanity.com/paper/1707.09378/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Jul 2017.

## Built on by (most cited first, 3 of 3)

- [A Topological Perspective on Causal Inference](https://mlsanity.com/paper/2107.08558/) (2021)
- [Are Bayesian networks typically faithful?](https://mlsanity.com/paper/2410.16004/) (2024)
- [Cross-Entropy Risk Estimation for Language Models: Inconsistency Must Be Dense, and the Holdout Method Is No Exception](https://mlsanity.com/paper/2608.15798/) (2026)

## Abstract (arXiv)

Topological models of empirical and formal inquiry are increasingly prevalent. They have emerged in such diverse fields as domain theory [1, 16], formal learning theory [18], epistemology and philosophy of science [10, 15, 8, 9, 2], statistics [6, 7] and modal logic [17, 4]. In those applications, open sets are typically interpreted as hypotheses deductively verifiable by true propositional information that rules out relevant possibilities. However, in statistical data analysis, one routinely receives random samples logically compatible with every statistical hypothesis. We bridge the gap between propositional and statistical data by solving for the unique topology on probability measures in which the open sets are exactly the statistically verifiable hypotheses. Furthermore, we extend that result to a topological characterization of learnability in the limit from statistical data.